The Remainder and Factor Theorems The Remainder Theorem

  • Slides: 15
Download presentation
The Remainder and Factor Theorems

The Remainder and Factor Theorems

The Remainder Theorem If a polynomial f(x) is divided by (x – a), the

The Remainder Theorem If a polynomial f(x) is divided by (x – a), the remainder is the constant f(a), and f(x) = q(x) ∙ (x – a) + f(a) where q(x) is a polynomial with degree one less than the degree of f(x). Dividend equals quotient times divisor plus remainder.

The Remainder Theorem Find f(3) for the following polynomial function. f(x) = 5 x

The Remainder Theorem Find f(3) for the following polynomial function. f(x) = 5 x 2 – 4 x + 3 f(3) = 5(3)2 – 4(3) + 3 f(3) = 5 ∙ 9 – 12 + 3 f(3) = 45 – 12 + 3 f(3) = 36

The Remainder Theorem Now divide the same polynomial by (x – 3). 5 x

The Remainder Theorem Now divide the same polynomial by (x – 3). 5 x 2 – 4 x + 3 3 5 – 4 3 15 33 5 11 36

The Remainder Theorem f(x) = 5 x 2 – 4 x + 3 f(3)

The Remainder Theorem f(x) = 5 x 2 – 4 x + 3 f(3) = 5(3)2 – 4(3) + 3 f(3) = 5 ∙ 9 – 12 + 3 f(3) = 45 – 12 + 3 5 x 2 – 4 x + 3 3 5 – 4 15 5 11 3 33 36 f(3) = 36 Notice that the value obtained when evaluating the function at f(3) and the value of the remainder when dividing the polynomial by x – 3 are the same. Dividend equals quotient times divisor plus remainder. 5 x 2 – 4 x + 3 = (5 x + 11) ∙ (x – 3) + 36

The Remainder Theorem Use synthetic substitution to find g(4) for the following function. f(x)

The Remainder Theorem Use synthetic substitution to find g(4) for the following function. f(x) = 5 x 4 – 13 x 3 – 14 x 2 – 47 x + 1 4 5 5 – 13 – 14 20 28 7 14 – 47 56 9 1 36 37

The Remainder Theorem Synthetic Substitution – using synthetic division to evaluate a function This

The Remainder Theorem Synthetic Substitution – using synthetic division to evaluate a function This is especially helpful for polynomials with degree greater than 2.

The Remainder Theorem Use synthetic substitution to find g(– 2) for the following function.

The Remainder Theorem Use synthetic substitution to find g(– 2) for the following function. f(x) = 5 x 4 – 13 x 3 – 14 x 2 – 47 x + 1 – 2 5 5 – 13 – 14 – 47 1 – 10 46 – 64 222 – 23 32 – 111 223

The Remainder Theorem Use synthetic substitution to find c(4) for the following function. c(x)

The Remainder Theorem Use synthetic substitution to find c(4) for the following function. c(x) = 2 x 4 – 4 x 3 – 7 x 2 – 13 x – 10 4 2 – 4 8 2 4 – 7 – 13 – 10 16 36 92 9 23 82

The Factor Theorem The binomial (x – a) is a factor of the polynomial

The Factor Theorem The binomial (x – a) is a factor of the polynomial f(x) if and only if f(a) = 0.

The Factor Theorem When a polynomial is divided by one of its binomial factors,

The Factor Theorem When a polynomial is divided by one of its binomial factors, the quotient is called a depressed polynomial. If the remainder (last number in a depressed polynomial) is zero, that means f(#) = 0. This also means that the divisor resulting in a remainder of zero is a factor of the polynomial.

The Factor Theorem x 3 + 4 x 2 – 15 x – 18

The Factor Theorem x 3 + 4 x 2 – 15 x – 18 x– 3 3 1 1 4 3 7 – 15 – 18 21 18 6 0 Since the remainder is zero, (x – 3) is a factor of x 3 + 4 x 2 – 15 x – 18. This also allows us to find the remaining factors of the polynomial by factoring the depressed polynomial.

The Factor Theorem x 3 + 4 x 2 – 15 x – 18

The Factor Theorem x 3 + 4 x 2 – 15 x – 18 x– 3 3 1 1 4 3 7 – 15 – 18 21 18 6 0 x 2 + 7 x + 6 (x + 6)(x + 1) The factors of x 3 + 4 x 2 – 15 x – 18 are (x – 3)(x + 6)(x + 1).

The Factor Theorem (x – 3)(x + 6)(x + 1). Compare the factors of

The Factor Theorem (x – 3)(x + 6)(x + 1). Compare the factors of the polynomials to the zeros as seen on the graph of x 3 + 4 x 2 – 15 x – 18.

The Factor Theorem Given a polynomial and one of its factors, find the remaining

The Factor Theorem Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. 1. x 3 – 11 x 2 + 14 x + 80 x– 8 2. 2 x 3 + 7 x 2 – 33 x – 18 x+6 (x – 8)(x – 5)(x + 2) (x + 6)(2 x + 1)(x – 3)